# The zipper macro and zipper moves

I continue from the post Generating set of Reidemeister moves for graphic lambda crossings , where the “crossing macros” over graphic lambda calculus were discussed.

Another interesting macro  (over graphic lambda calculus) is the zipper, together with its associated zipper moves.

Let’s take $n \geq 2$ a natural number and let’s consider the following graph in $GRAPH$, called the n-zipper:

At the left is the n-zipper graph; at the right is a NOTATION for it, or a macro.  We could as well take $n =2$, with obvious modifications of the figure, so the 2-zipper exists. Even $n=1$ makes sense, but the 1-zipper is kind of degenerate, see later.

There is a graphic beta move which we can perform on the n-zipper graph. In the following picture I figured in red the place where the graphic beta move is applied.

In terms of zipper notation this graphic beta move has the following appearance:

We see that a n-zipper transforms into a (n-1)-zipper plus an arrow. We may repeat this move, as long as we can. What is the result? A n-zipper move:

The 1-zipper move, called $ZIP_{1}$ is just the graphic beta move, which transforms the 1-zipper into two arrows.

Nice, now what can we do with zippers and crossings?